Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{x^2 + 12x + 27}{x^2 + 7x + 12}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 12x + 27}{x^2 + 7x + 12} = \dfrac{(x + 9)(x + 3)}{(x + 4)(x + 3)} $ Notice that the term $(x + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 3)$ gives: $k = \dfrac{x + 9}{x + 4}$ Since we divided by $(x + 3)$, $x \neq -3$. $k = \dfrac{x + 9}{x + 4}; \space x \neq -3$